Periodic and preperiodic points of the Mandelbrot set can be found by means of
polynomials gn(c) = fcon(0) . A few of
these polynomials are plotted below near the preperiodic point
mo = -2 (the boundary crisis point).
As we know in the vicinity of any preperiodic point the Mandelbrot set is
self-similar. Let at the crisis point c = -2 +
e /rn .
H.Hurwitz et al. have shown that for
-> 2 cos(e 1/2)
where rn = 4n/6 and
n > 1 . A few of the polynomials gn(-2 +
are plotted as a function of e in Fig.2.
As n increases (5, 6, 7...) the polynomials approach the
universal function 2 cos(e 1/2) .
The range - in e - of good approximation
rapidly increases with increasing n.
The superstable values of c (near -2 for n sufficiently
large) are associated with the roots of
cos(e 1/2). We denote the real roots
of gn by cn,j where j = 1
corresponds to the root closest to c = -2 with the kneading sequence
CLRn-2, j = 2, to the next closest with kneading
sequence CLRn-3L and so on
cn,j = -2 + 6p
2(j - 1/2)2 / 4n .
For n = 25 the cosine approximation yields more then 10,000
real roots closest to c = -2 with an error of less then 1%.
One can get an approximate formula for locations of preperiodic points too.
The generalisation of universal scaling constants for these M-set copies
16 n/[6p 2
(2j - 1)2] ,
4 n/[2p (2j - 1)] .
 H.Hurwitz, M.Frame, D.Peak "Scaling symmetries in nonlinear
dynamics. A view from parameter space" Physica D 81 (1995) 23.
Tip of the Mandelbrot set
The M-set near the preperiodic point
mo = -2 is self-similar under scaling by a factor of
|l| = 4 , therefore every next
CLRn (n = 1,2...) orbit is 4 times closer to
mo . Corresponding miniature M-set shrinks by a factor of
l2 and eventually disappear.
They converge to mo with symbolic dynamics [CL]R
where [CL] is preperiodic "tail" of the point and R
is its periodic part. The point is the tip of the Mandelbrot set antenna.
Period-n orbit CLRn for large n gets near
unstable period-1 orbit, leaves it and eventually returns into the z = 0
point. Period-4 and period-5 orbits CLR2 and
CLR3 are shown here. Preperiodic point with the pattern
[CLR3]L and c = -1.97393 lies between these two
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updated 29 October 2002